# Comprehensive Exam Preparation

This information is designed to help mathematics and statistics graduate students prepare for the comprehensive exams (Comps). These exams are tough, and all too frequently, good students turn in mediocre papers or even fail. Our philosophy is that with better information, the whole process will be less painful, more successful, and ultimately, more beneficial for the students.

Studying for the Comps is a lot of work, but it can be a rewarding experience, both in the short term and in the long term. And your Masters degree will be worth much more if you really learn the material thoroughly, rather than just doing the minimum to scrape by.

## Benefits

- No matter what your plans are, you came to CSULB to learn more math or statistics, and you'll spend several semesters learning new material. The Comps, firstly, give you a chance to review what you've learned after the semester is over and organize an entire subject coherently in your mind. Secondly, the rules force you to master two different subjects at once. For the rest of your life, you'll remember that for at least one week, you had a command of a broad spectrum of mathematics. It's a good feeling, and it will give you the confidence that you can always retrieve that knowledge even if you forget details later.
- If you're heading for a job in industry, you'll be the expert in your field.
- If you're looking at academic jobs in high schools or community colleges, you will be a much better teacher if you understand the deeper math underlying the ideas you're teaching. You will become the expert to whom your colleagues turn for help.
- If you're looking at doctoral programs, you probably know that almost all programs have a rigorous battery of written exams that must be passed in the first couple of years. The Comps are excellent practice for these exams.
- If you get A's on both your Comps, then you are officially designated by the department to have graduated with Distinction. This is something you can put on your CV/resume and show to employers, etc.
- Even if you don't graduate with Distinction, you can brag about good grades on your Comps.
- Professors grading your Comps will be impressed by strong papers. Many will be willing to express their approval in formal recommendation letters. Even without formal recommendations, the academic community is small, and it is nice to know that your work is well respected around CSULB.

## Timeline

You'll see fliers in the department at the end of each semester announcing signups for the Comps. Does that mean it's time to sign up and hit the books? Frankly, you should have started well before then. The exams are administered at the beginning of the following semester, and good preparation takes much longer than a few weeks during the break. Besides, professors are hard to track down during the break, and one of your best resources is asking professors for help.

After you take the graduate class that corresponds to a particular exam, it takes about one more semester to prepare thoroughly for the Comps. You'll need to organize the material in your own way, let it sink in, and practice applying it to solve problems. In many fields, you can do this as you take the second semester of the graduate class. Even if that second-semester material is not covered directly on the Comps, learning it will reinforce the concepts you learned in the first semester and give you practice applying that material in context.

Task | Spring Exam of Next Year | Fall Exam of Next Year |
---|---|---|

Take first semester graduate class | Spring of this year | Fall of this year |

Organize notes, learn theorems, practice problems, consult with other scholars | Fall of this year | Spring of this year |

Take second semester class, if available | Fall of this year | Spring of this year |

Sign up for exam | December | May |

Final review | January | Summer |

Take the exam! | February | September |

## Subject-Specific Help

The following will help you with your exam preparations.

## Pure Math Comps

These exams are for pure math graduate program.

This Comp primarily covers material from the following course:

- MATH 540

However, it also covers some material from the following courses:

- MATH 247
- MATH 347
- MATH 545

### Sample Comps

### Formal Syllabus

Algebra Comp syllabus (PDF) - includes topics and reference materials.

### Informal Advice from Faculty

Ok, let's look at the old exams. What can we learn?

**Partial answers matter.**You always choose six problems. That's the first mistake people make: If you only answer five, you're throwing away points. A partial answer on the sixth is better than nothing. If you're worried about looking foolish, you can always write a note: "I have proved that this homomorphism is injective. I don't know how to prove that it is surjective, but I believe that it is." (Of course, don't write garbage just to get points -- the graders will distinguish between irrelevant rambling and steps that contribute to a correct solution.)**Don't just study group theory.**Most students feel most confident about group theory. But it will probably be impossible for you to choose six problems on group theory. Either there won't be six problems given, or the directions will force you to choose some from ring theory and linear algebra. Some students just study group theory and hope to pass on that. This is, in theory, possible: Absolutely perfect scores on four problems on group theory would, in some years, be enough to pass the exam. But that's very, very unlikely. First, enough group theory problems are tricky enough that it will be very hard to get perfect scores. Second, 66% isn't guaranteed to pass, and even if it does, it would probably only be a C (especially since graders would look askance at a paper that only answered group theory questions).*Bottom line:*You should learn some ring theory and linear algebra too.

**Linear algebra is the overlooked low-hanging fruit of the exam.**Every recent exam has had a linear algebra problem on it, and the Spring 06 exam had two. Most people didn't try them. The students who did picked up big points on them. That's because they're easier than the other topics: They cover undergraduate linear algebra (MATH 247 and maybe some 347) as opposed to the other topics, which cover graduate algebra. So review your notes from MATH 247 and/or 347 and grab some easy points.**Give extra practice to certain topics.**To be safe, you should learn all the topics on the syllabus. But certain topics seem to come up on almost every exam, so these topics are definitely worthy of extra practice:- Groups
- Counting problems in which you are given a group of order n (some fixed number) and asked to find a subgroup or element of order k. This, of course, uses Sylow Theory.
- Classification of all abelian groups of order n.
- Showing that a subgroup is normal. There may be many different strategies for this.
- Common examples: symmetric groups (Sn), alternating groups (An), dihedral groups (Dn), and cyclic groups (Zn). Even if they aren't mentioned in the questions, you may need to use them as examples in your answers.

- Rings
- Ideals and quotient rings.
- Prime and maximal ideals (or quotient rings being integral domains or fields, which is the same idea).
- Examples of the form Z (integers) adjoin the square root of a negative number, like -1 (Z[i]), -3, or -5. These examples often come up in connection with Euclidean domains, principal ideal domains, and unique factorization domains. You should know the implications and counterexamples between these ideas. Other related concepts are prime and irreducible elements.

- Linear Algebra
- Matrices.
- Vector spaces, basis, and dimension.
- Eigenvalues and eigenspaces.

- Groups

### Faculty Who Can Help

The following faculty members are knowledgeable about this field and are willing to answer questions from students preparing for the Algebra Comp:

- Dr. John Brevik
- Dr. Will Murray
- Dr. Joshua Sack
- Dr. Robert Valentini

This Comp primarily covers material from the following course:

- MATH 562A

However, it also covers some material from the following courses:

- MATH 361B
- MATH 461
- MATH 562B

### Sample Comps

Here are some sample Comps in this field:

- Complex Analysis Comp - Spring 2021 (PDF)
- Complex Analysis Comp - Spring 2019 (PDF)
- Complex Analysis Comp - Fall 2018 (PDF)

### Formal Syllabus

Complex Analysis Comp syllabus (PDF) - includes topics and reference materials.

### Faculty Who Can Help

The following faculty members are knowledgeable about this field and are willing to answer questions from students preparing for the Complex Analysis Comp:

- Dr. Joseph Bennish
- Dr. Ngo Viet

This Comp primarily covers material from the following course:

- MATH 561A

However, it also covers some material from the following courses:

- MATH 361B
- MATH 561B

### Sample Comps

Here are some sample Comps in this field:

- Real Analysis Comp - Spring 2021 (PDF)
- Real Analysis Comp - Spring 2020 (PDF)
- Real Analysis Comp - Fall 2019 (PDF)

### Formal Syllabus

Real Analysis Comp syllabus (PDF) - includes topics and reference materials.

### Informal Advice from Faculty

- There are several different standard books for real analysis (see the Syllabus for references), and they develop key topics in different orders. For example, some prove the Monotone Convergence Theorem from scratch and then derive Fatou's Lemma as a consequence, and some do it the other way around.
- If you are asked to prove one of these seminal facts on the exam, you may do it either way, but you should define clearly the terms you will use and give careful statements of theorems you plan to use without proof. To be safe, you may want to use the development followed in your version of MATH 561A. The most recent teacher of 561A is always one of the graders for Real Analysis, so (s)he will be familiar with the development you learned and will know if you are following it correctly.

### Faculty Who Can Help

The following faculty members are knowledgeable about this field and are willing to answer questions from students preparing for the Real Analysis Comp:

- Dr. Joseph Bennish
- Dr. Florence Newberger

This Comp primarily covers material from the following course:

- MATH 550

However, it also covers some material from the following courses:

- MATH 361B
- MATH 555

### Sample Comps

Here are some sample Comps in this field:

### Formal Syllabus

Topology Comp syllabus (PDF) - includes topics and reference materials.

### Informal Advice from Faculty

For a complete set of course materials for a past MATH 550 class, please see Dr. Ryan Blair's MATH 550 lecture notes.

### Faculty Who Can Help

The following faculty members are knowledgeable about this field and are willing to answer questions from students preparing for the Topology Comp:

- Dr. Ryan Blair
- Dr. Yu Ding
- Dr. David Gau
- Dr. Joshua Sack

## Applied Math Comps

These exams are for applied math graduate program.

This Comp primarily covers material from the following course:

- MATH 564

However, it also covers some material from the following course:

- MATH 361B
- MATH 364A

### Sample Comps

Here are some sample Comps in this field:

- Ordinary Differential Equations - Sample 1 (PDF)
- Ordinary Differential Equations - Sample 2 (PDF)
- Ordinary Differential Equations - Sample 3 (PDF)

### Formal Syllabus

Ordinary Differential Equations Comp syllabus (PDF) - includes topics and reference materials.

### Faculty Who Can Help

The following faculty member is knowledgeable about this field and is willing to answer questions from students preparing for the Ordinary Differential Equations Comp:

- Dr. Eun Heui Kim

This Comp primarily covers material from the following course:

- MATH 576

However, it also covers some material from the following courses:

- MATH 323
- MATH 361B
- MATH 364B

### Sample Comps

Here are some sample Comps in this field:

- Numerical Analysis Comp - Spring 2006 (PDF)
- Numerical Analysis Comp - Fall 2006 (PDF)
- Numerical Analysis Comp - Spring 2007 (PDF)

### Formal Syllabus

Numerical Analysis Comp syllabus (PDF) - includes topics and reference materials.

### Faculty Who Can Help

The following faculty members are knowledgeable about this field and are willing to answer questions from students preparing for the Numerical Analysis Comp:

- Dr. Bruce Chaderjian
- Dr. Tangan Gao
- Dr. Wen-Qing Xu

This Comp primarily covers material from the following course:

- MATH 570

However, it also covers some material from the following courses:

- MATH 364A
- MATH 463
- MATH 470

### Sample Comps

Here are some sample Comps in this field:

- Partial Differential Equations Comp - Spring 2022 (PDF)
- Partial Differential Equations Comp - Fall 2020 (PDF)
- Partial Differential Equations Comp - Fall 2017 (PDF)

You're also given a page of formulas to use during the Partial Differential Equations Comp (PDF).

### Formal Syllabus

Partial Differential Equations Comp syllabus (PDF) - includes topics and reference materials.

## Math Education Comps

These exams are for math education graduate program.

This Comp primarily covers material from the following courses:

- MTED 511
- MTED 512

### Sample Comp

Here is a sample Comp in this field:

- Math Education Core/Elective Comp Samples (DOCX) - revised 2024

### Formal Syllabus

Mathematics Education Comprehensive Exam Syllabi (PDF) - revised 2024, includes topics and reference materials.

### Faculty Who Can Help

The following faculty members are knowledgeable about this field and are willing to answer questions from students preparing for the Math Education Core Comp:

- Dr. Babette Benken
- Dr. Joshua Chesler
- Dr. Brian Katz
- Dr. Xuhui Li
- Dr. Antonio Martinez
- Dr. Jeffrey Pair

These Comps cover material from the following courses:

- Algebra: MTED 540
- Geometry: MTED 550
- Analysis: MTED 560
- Mathematical Modeling: MTED 570
- Probability and Statistics: MTED 580

However, they also cover some material from the following courses:

- MTED 511
- MTED 512
- MATH 361A (for the Analysis Comp, MTED 560)

### Sample Comp

Here is a sample Comp in this field:

- Math Education Core/Elective Comp Samples (DOCX) - revised 2024

### Formal Syllabus

Mathematics Education Comprehensive Exam Syllabi (PDF) - revised 2024, includes topics and reference materials.

### Faculty Who Can Help

The following faculty members are knowledgeable about this field and are willing to answer questions from students preparing for the Math Education Electives Comp:

- Dr. Babette Benken
- Dr. Joshua Chesler
- Dr. Brian Katz
- Dr. Xuhui Li
- Dr. Antonio Martinez
- Dr. Jeffrey Pair

## Applied Statistics Comps

These exams are for applied statistics graduate program.

This Comp primarily covers material from the following course:

- STAT 530

However, it also covers some material from the following course(s):

- STAT 381

### Sample Comps

Here are some sample Comps in this field:

- Experimental Design and Analysis Comp - Fall 2021 (PDF)
- Experimental Design and Analysis Comp - Spring 2013 (PDF)
- Experimental Design and Analysis Comp - Spring 2012 (PDF)

You're also given a page of formulas to use during the Experimental Design and Analysis Comp (PDF).

### Formal Syllabus

Experimental Design and Analysis Comp syllabus (PDF) - includes topics and reference materials.

### Faculty Who Can Help

The following faculty members are knowledgeable about this field and are willing to answer questions from students preparing for the Experimental Design and Analysis Comp:

- Dr. Sung Kim
- Dr. Olga Korosteleva
- Dr. Tianni Zhou

This Comp primarily covers material from the following course:

- STAT 520

However, it also covers some material from the following course:

- STAT 381

### Sample Comps

Here are some sample Comps in this field:

- Statistical Inference Comp - Fall 2015 (PDF)
- Statistical Inference Comp - Spring 2015 (PDF)
- Statistical Inference Comp - Fall 2014 (PDF)

### Formal Syllabus

Statistical Inference Comp syllabus (PDF). Includes topics and reference materials.

### Faculty Who Can Help

The following faculty members are knowledgeable about this field and are willing to answer questions from students preparing for the Statistical Inference Comp:

- Dr. Yonghee Kim-Park
- Dr. Hojin Moon
- Dr. Kagba Suaray

## Mistakes to Avoid

Over the years we've had many good students perform poorly on the Comps because they made the following mistakes. Don't make them yourself!

- Don't take the Comp before you've finished the second semester graduate class if the class spans two semesters (Algebra, Complex Analysis, Functional Analysis, Real Analysis, Topology). Lots of people think that since the Comp primarily covers first semester material, it should be taken as soon as that semester is finished. In reality, during the second semester you reinforce and apply what you learned in the first semester. Concepts that were new and confusing at the end of the first semester will be easy and natural by the end of the second.
- Study for
*BOTH*exams well in advance. Remember that you take them on two consecutive weekends. Some people put off studying for the second Comp until after the first, giving themselves one week to study for the second. That's not nearly enough time. - Don't take the Comps once before you're ready, just to get the feel for them. You only get two chances. If you burn the first one without serious studying, you're putting yourself under tremendous pressure for the second.
- Don't assume the Comp will be like the final of the relevant grad class. First, the Comp is usually harder. Second, the topics may vary a little. Check out the subject-specific help.
- Work through old Comps. Problems occasionally get repeated, and broad methods and themes often get repeated. Also, sometimes you are asked to prove classic theorems, so it may be worth memorizing proofs of these, as long as you understand what you are memorizing.
- But don't expect to pass on memorization. Although problems occasionally get repeated verbatim, more often the writers create new problems requiring you to apply the same basic principles in slightly different situations. It is pretty easy for a grader to tell when a student is regurgitating a verbatim answer to the wrong problem, and you don't get much credit for it. If you don't understand what you're memorizing, you're in trouble. To test your understanding, try modifying the parameters of the problem a little, or ask yourself to what other situations the same principles would apply.
- Talk to other students or professors about the Comps. Don't study in complete isolation. The more you work problems and check your solutions with other students and professors, the better.
- Answer as many problems as you're allowed to, even if some of your answers are incomplete. On the Algebra Comp, for example, you normally choose six problems. If you only answer five, you're throwing away points. A partial answer on the sixth is better than nothing. If you're worried about looking foolish, you can always write a note: "I recognize that this answers Part X, but not Part Y, of the problem." (Of course, don't write garbage just to get points -- the graders will distinguish between irrelevant rambling and steps that contribute to a correct solution.)